Linear Algebra and Its Applications, Exercise 2.2.6

Exercise 2.2.6. Consider the following system of linear equations:

$\begin{bmatrix} 1&2&2 \\ 2&4&5 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$

Find the general solution expressed as the sum of a particular solution to $Ax = b$ and the solution to $Ax = 0$ .

Answer: We perform elimination by subtracting 2 times the first row from the second row:

$\begin{bmatrix} 1&2&2 \\ 2&4&5 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&2&2 \\ 0&0&1 \end{bmatrix}$

This completes elimination. The matrix $U$ has pivots in columns 1 and 3, so that the basic variables are $u$ and $w$ and the free variable is $v$ .

We can replace the system $Ax = 0$ with the new system $Ux = 0$ :

$\begin{bmatrix} 1&2&2 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

From the second equation we have $w = 0$ . Substituting into the first equation gives us $u +2v + 2w = u + 2v = 0$ or $u = -2v$ . The general solution to $Ax = 0$ is thus

$x = \begin{bmatrix} -2v \\ v \\ 0 \end{bmatrix} = v \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}$

We now consider the inhomogeneous system $Ax = b$ or

$\begin{bmatrix} 1&2&2 \\ 2&4&5 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$

The elimination sequence from above produces the system $Ux = c$ or

$\begin{bmatrix} 1&2&2 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

From the second equation we have $w = 2$ . Substituting into the first equation and setting the free variable $v = 0$ produces $u + 2v + 2w = u + 0 + 4 = 1$ or $u = -3$ . The particular solution is thus $x = (-3, 0, 2)$ .

We can combine the particular solution to this system with the general solution to $Ax = 0$ to produce the general solution $x$ for the system $Ax = b$ :

$x = \begin{bmatrix} -3 \\ 0 \\ 2 \end{bmatrix} + v \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

2 Responses to Linear Algebra and Its Applications, Exercise 2.2.6

Nate says:

September 27, 2011 at 4:24 am

you solved the particular solution right but copied it wrong in your total solution

- hecker says:
  
  September 27, 2011 at 1:45 pm
  
  Thanks for catching this! I’ve updated the post to fix it.